big o statement prove or disprove (impossible)

Question

This question is harder than it looks folks

for all a in the reals and for all b in the reals, [(a <= b) => (n^a is O(n^b))]

n^a is O(n^b) if n^a <= cn^b for some n>= n, (n less than or equal to n knot)

then n^a/n^b <= c

n^a-b <= c

what would my c and n knot be??

Answer 1

If $n\ge 1$, we have $$\frac{n^a}{n^b}=\frac{1}{n^{b-a}}\le 1.$$ Thus we can take $n_0=1$ and $c=1$.